Page 272 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

SYSTEM IDENTIFICATION 261

techniques have been developed. For linear systems, processing in the
frequency domain may be advantageous. Another possibility is to pro-
cess the measurements sequentially. The trick is to regard the parameters
as state vectors a(i). Static parameters do not change in time. So, the
corresponding state equation is a(i þ 1) ¼ a(i). Sometimes, it is useful to
allow slow variations in a(i). This is helpful in order to model drift
phenomena, but also to improve the convergence properties of the
procedure. A simple model would be a process that is similar to random
walk (Section 4.2.1): a(i þ 1) ¼ a(i) þ w(i). The white noise sequence
w(i) is the driving force for the changes. Its covariance matrix C w should
be small in order to prevent a too wild behaviour of a(i).
Using this model, equation (8.6) transforms into:

" # " #
xði þ 1Þ fðxðiÞ; wðiÞ; aðiÞÞ
¼ (state equation)
aði þ 1Þ aðiÞþ wðiÞ ð8:7Þ
zðiÞ¼ hðxðiÞ; vðiÞÞ (measurement equation)

The original state vector x(i) has been augmented with a(i). The new
state equation can be written as x(i þ 1) ¼ f(x(i),w(i), w(i)) with
def T
T
T
x(i) ¼ [ x (i) a (i) ] . This opens the door to simultaneous online
estimation of both x(i)and a(i) using the techniques discussed in
Chapter 4. However, the new state function f( ) is nonlinear and for
online estimation we must resort to estimators that can handle these
nonlinearities, e.g. extended Kalman filtering (Section 4.2.2), or particle
filtering (Section 4.4).
Note that if w(i) 0, then a(i) is a random constant and (hopefully) its
a
estimate ^ a(i) converges to a constant. If we allow w(i) to deviate from zero
by setting C w to some (small) nonzero diagonal matrix, the estimator
becomes adaptive. It has the potential to keep track of parameters that drift.

Example 8.3 Parameter estimation for the hydraulic system
In order to estimate the parameters of the three models of the
hydraulic system using the data from the previous example the
following procedure was applied. First, a particle filter was executed
in order to get a rough indication of the magnitudes of the parameters.
Figure 8.4(a) shows the results of the filter applied to the Torricelli
model. In this model, there are two parameters A 1 and A 2 which were
2
both initiated with a uniform distribution between 0 and 2(cm ). The
parameters were modelled with A(i þ 1) ¼ A(i) þ !(i) where !(i)is
2
white noise with a standard deviation of 0:004(cm ).